Bivariate multiple orthogonal polynomials of mixed type on the step-line

TL;DR

This paper investigates bivariate multiple orthogonal polynomials of mixed type on the step-line, deriving their properties, recurrence relations, and kernels, with an application to Jacobi-Pi eiro polynomials.

Contribution

It introduces a novel analysis framework for these polynomials using LU factorization and establishes new relations and formulas, including an ABC-type theorem.

Findings

Derived orthogonality and biorthogonality relations.

Established recurrence relations and Christoffel-Darboux formulas.

Computed Jacobi-Pi eiro polynomials via LU factorization.

Abstract

This article studies bivariate multiple orthogonal polynomials of the mixed type on the step-line. The analysis is based on the LU factorization of a moment matrix specifically adapted to this framework. The orthogonality and biorthogonality relations satisfied by these polynomials are identified, and their precise multi-degrees are determined. The corresponding recurrence relations and the growing band matrices that encode them are also derived. Christoffel-Darboux kernels and the associated Christoffel-Darboux-type formulas are obtained. An ABC-type theorem is established, relating the inverse of the truncated moment matrix to these kernels. As an illustration, the bivariate Jacobi-Pi\~neiro multiple orthogonal polynomials of mixed type on the triangle are computed by means of an LU factorization implemented in a dedicated Maple script.